T-9

Goal: the goal of this problem is to determine when the solution of the distributional equations corresponding to localization is unstable, providing an estimate of thee mobility edge on the Bethe lattice.
Techniques: stability analysis, Laplace transforms.

Problems

In this Problem we will determine for which values of parameters localization is stable, estimating the critical value of disorder where the transition to a delocalised phase occurs. Recall the results of Problem 8: the real and imaginary parts of the local self energy satisfy the self-consistent equations:

$\Gamma _{a}=\sum _{b\in \partial a}V_{ab}^{2}{\frac {\Gamma _{b}+\eta }{(E-\epsilon _{b}-R_{b})^{2}+(\Gamma _{b}+\eta )^{2}}},\quad \quad R_{a}=\sum _{b\in \partial a}V_{ab}^{2}{\frac {E-\epsilon _{b}-R_{b}}{(E-\epsilon _{b}-R_{b})^{2}+(\Gamma _{b}+\eta )^{2}}}$

These equations admit the solution $\Gamma _{a}=\Gamma _{b}=0$ when $\eta =0$ , which corresponds to localization. We now determine when this solution becomes unstable.

Problem 9: an estimate of the mobility edge

Imaginary approximation and distributional equation. We consider the equations for $\Gamma _{a}$ and neglect the terms $R_{b}$ in the denominators, which couple the equations to those for the real parts of the self energies (“imaginary” approximation). Moreover, we assume to be in the localized phase, where $\Gamma _{a}\sim \eta \ll 1$ . Finally, we set $V_{ab}\equiv V$ and $E=0$ for simplicity. Show that under these assumptions the probability density for the imaginary part, $P_{\Gamma }(\Gamma )$ , satisfies
$P_{\Gamma }(\Gamma )=\int \prod _{b=1}^{k}d\epsilon _{b}\,p(\epsilon _{b})\int \prod _{b=1}^{k}d\Gamma _{b}\,P_{\Gamma }(\Gamma _{b})\delta \left(\Gamma -V^{2}\sum _{b\in \partial a}{\frac {\Gamma _{b}+\eta }{\epsilon _{b}^{2}}}\right)$

Show that the Laplace transform of this distribution, $\Phi (s)=\int _{0}^{\infty }d\Gamma e^{-s\Gamma }P_{\Gamma }(\Gamma )$ , satisfies

$\Phi (s)=\left[\int d\epsilon \,p(\epsilon )e^{-{\frac {sV^{2}\eta }{\epsilon ^{2}}}}\Phi \left({\frac {sV^{2}}{\epsilon ^{2}}}\right)\right]^{k}$

The stability analysis. We now assume to be in the localized phase, when for $\eta \to 0$ $\eta \to 0$ the distribution $P_{\Gamma }(\Gamma )\to \delta (\Gamma )$ $P_{\Gamma }(\Gamma )\to \delta (\Gamma )$ . We wish to check the stability of our assumption. This is done by controlling the tails of the distribution $P_{\Gamma }(\Gamma )$ $P_{\Gamma }(\Gamma )$ for finite $\eta$ $\eta$ .
- For finite $\eta$ , we expect that typically $\Gamma _{a}\sim \eta \ll 1$ , and thus $P_{\Gamma }(\Gamma )$ should have a peak at this scale; however, we also expect some power law decay $P_{\Gamma }(\Gamma )\sim \Gamma ^{-\alpha }$ for large $\Gamma$ . These tails are contributed by the events in which one of the descendants $b$ is such that $\epsilon _{b}$ is very small, in which case $\Gamma _{a}\sim 1/\epsilon _{b}^{2}\gg 1$ . Show that if $\Gamma \sim 1/\epsilon ^{2}$ and $p(\epsilon )$ is not gapped around zero, then $P_{\Gamma }(\Gamma )\sim \Gamma ^{-3/2}$ , i.e. the distribution has tails contributed by these events in which the local random potential happen to be very small.
- Assume more generally that $P_{\Gamma }(\Gamma )\sim \Gamma ^{-\alpha }$ for large $\Gamma$ and $\alpha \in (1,3/2]$ . Show, using a dimensional analysis argument, that this corresponds to a non-analytic behaviour of the Laplace transform, $\Phi (s)\sim 1-A|s|^{\beta }$ for $s$ small, with $\beta =\alpha -1\in (0,1/2]$ .
- Show that the equation for $\Phi (s)$ gives for $s$ small $1-As^{\beta }=1-Ak\int d\epsilon \,p(\epsilon ){\frac {s^{\beta }V^{2\beta }}{\epsilon ^{2\beta }}}+o(s^{\beta })$ , and therefore this is consistent provided that there exists a $\beta \in (0,1/2]$ solving
  $1=k\int d\epsilon \,p(\epsilon )\left({\frac {V}{|\epsilon |}}\right)^{2\beta }\equiv kI(\beta ).$

Behaviour of the integral

I(\beta )

in the case of uniformily distributed disorder, for

W<W_{c}

.

The critical disorder. Consider now local fields $\epsilon$ taken from a uniform distribution in $[-W/2,W/2]$ . Compute $I(\beta )$ and show that it is non monotonic, with a local minimum $\beta ^{*}$ in the interval $[0,1/2]$ . Show that $I(\beta ^{*})$ increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when $I(\beta ^{*})=k^{-1}$ . Show that this gives the following estimate for the critical disorder at which the transition to delocalisation occurs:
${\frac {W_{c}}{V}}=\,2ke\log \left({\frac {W_{c}}{2V}}\right)\sim \,2e\,k\log \left(k\right)$

Why the critical disorder increases with $k$ ?